One can prove this alternatively by contradiction. Namely, supposing toward contradiction that one has a finite list of all the primes p1, …, pn, one then multiplies them together and adds one N = p1p2⋯pn + 1. This new number is not a multiple of any pi, and so its prime factorization must involve new primes, not on the list. This contradiction our assumption that we had all the prime numbers on the list.
